Quake 1 is the result of the team being split on what direction to take the next game after Doom II/Final Doom; half wanted to make a
Lovecraftian 3D FPS, whilst the rest just wanted to do another straight-up FPS, some internal issues occurred and when the deadline loomed they simply mashed the assets they had together and the rest is history.
Quake III Arena was indeed a competitive multiplayer FPS, whose level design very much revolved around an "Arena" style and the game played as much too - Coincidentally it came out round about the same time as the 1st Unreal Tournament, which also followed the same concepts as Q3A - Copycats or just brilliant minds thinking alike?